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Mirrors > Home > MPE Home > Th. List > lidlrsppropd | Structured version Visualization version GIF version |
Description: The left ideals and ring span of a ring depend only on the ring components. Here 𝑊 is expected to be either 𝐵 (when closure is available) or V (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.) |
Ref | Expression |
---|---|
lidlpropd.1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
lidlpropd.2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
lidlpropd.3 | ⊢ (𝜑 → 𝐵 ⊆ 𝑊) |
lidlpropd.4 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
lidlpropd.5 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) ∈ 𝑊) |
lidlpropd.6 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
Ref | Expression |
---|---|
lidlrsppropd | ⊢ (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lidlpropd.1 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
2 | rlmbas 19016 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘(ringLMod‘𝐾)) | |
3 | 1, 2 | syl6eq 2660 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ringLMod‘𝐾))) |
4 | lidlpropd.2 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
5 | rlmbas 19016 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘(ringLMod‘𝐿)) | |
6 | 4, 5 | syl6eq 2660 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(ringLMod‘𝐿))) |
7 | lidlpropd.3 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝑊) | |
8 | lidlpropd.4 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) | |
9 | rlmplusg 19017 | . . . . . 6 ⊢ (+g‘𝐾) = (+g‘(ringLMod‘𝐾)) | |
10 | 9 | oveqi 6562 | . . . . 5 ⊢ (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘(ringLMod‘𝐾))𝑦) |
11 | rlmplusg 19017 | . . . . . 6 ⊢ (+g‘𝐿) = (+g‘(ringLMod‘𝐿)) | |
12 | 11 | oveqi 6562 | . . . . 5 ⊢ (𝑥(+g‘𝐿)𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦) |
13 | 8, 10, 12 | 3eqtr3g 2667 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘(ringLMod‘𝐾))𝑦) = (𝑥(+g‘(ringLMod‘𝐿))𝑦)) |
14 | rlmvsca 19023 | . . . . . 6 ⊢ (.r‘𝐾) = ( ·𝑠 ‘(ringLMod‘𝐾)) | |
15 | 14 | oveqi 6562 | . . . . 5 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐾))𝑦) |
16 | lidlpropd.5 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) ∈ 𝑊) | |
17 | 15, 16 | syl5eqelr 2693 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘(ringLMod‘𝐾))𝑦) ∈ 𝑊) |
18 | lidlpropd.6 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) | |
19 | rlmvsca 19023 | . . . . . 6 ⊢ (.r‘𝐿) = ( ·𝑠 ‘(ringLMod‘𝐿)) | |
20 | 19 | oveqi 6562 | . . . . 5 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦) |
21 | 18, 15, 20 | 3eqtr3g 2667 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘(ringLMod‘𝐾))𝑦) = (𝑥( ·𝑠 ‘(ringLMod‘𝐿))𝑦)) |
22 | baseid 15747 | . . . . . . 7 ⊢ Base = Slot (Base‘ndx) | |
23 | eqid 2610 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
24 | 22, 23 | strfvi 15741 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘( I ‘𝐾)) |
25 | rlmsca2 19022 | . . . . . . 7 ⊢ ( I ‘𝐾) = (Scalar‘(ringLMod‘𝐾)) | |
26 | 25 | fveq2i 6106 | . . . . . 6 ⊢ (Base‘( I ‘𝐾)) = (Base‘(Scalar‘(ringLMod‘𝐾))) |
27 | 24, 26 | eqtri 2632 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘(Scalar‘(ringLMod‘𝐾))) |
28 | 1, 27 | syl6eq 2660 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐾)))) |
29 | eqid 2610 | . . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
30 | 22, 29 | strfvi 15741 | . . . . . 6 ⊢ (Base‘𝐿) = (Base‘( I ‘𝐿)) |
31 | rlmsca2 19022 | . . . . . . 7 ⊢ ( I ‘𝐿) = (Scalar‘(ringLMod‘𝐿)) | |
32 | 31 | fveq2i 6106 | . . . . . 6 ⊢ (Base‘( I ‘𝐿)) = (Base‘(Scalar‘(ringLMod‘𝐿))) |
33 | 30, 32 | eqtri 2632 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘(Scalar‘(ringLMod‘𝐿))) |
34 | 4, 33 | syl6eq 2660 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(Scalar‘(ringLMod‘𝐿)))) |
35 | 3, 6, 7, 13, 17, 21, 28, 34 | lsspropd 18838 | . . 3 ⊢ (𝜑 → (LSubSp‘(ringLMod‘𝐾)) = (LSubSp‘(ringLMod‘𝐿))) |
36 | lidlval 19013 | . . 3 ⊢ (LIdeal‘𝐾) = (LSubSp‘(ringLMod‘𝐾)) | |
37 | lidlval 19013 | . . 3 ⊢ (LIdeal‘𝐿) = (LSubSp‘(ringLMod‘𝐿)) | |
38 | 35, 36, 37 | 3eqtr4g 2669 | . 2 ⊢ (𝜑 → (LIdeal‘𝐾) = (LIdeal‘𝐿)) |
39 | fvex 6113 | . . . . 5 ⊢ (ringLMod‘𝐾) ∈ V | |
40 | 39 | a1i 11 | . . . 4 ⊢ (𝜑 → (ringLMod‘𝐾) ∈ V) |
41 | fvex 6113 | . . . . 5 ⊢ (ringLMod‘𝐿) ∈ V | |
42 | 41 | a1i 11 | . . . 4 ⊢ (𝜑 → (ringLMod‘𝐿) ∈ V) |
43 | 3, 6, 7, 13, 17, 21, 28, 34, 40, 42 | lsppropd 18839 | . . 3 ⊢ (𝜑 → (LSpan‘(ringLMod‘𝐾)) = (LSpan‘(ringLMod‘𝐿))) |
44 | rspval 19014 | . . 3 ⊢ (RSpan‘𝐾) = (LSpan‘(ringLMod‘𝐾)) | |
45 | rspval 19014 | . . 3 ⊢ (RSpan‘𝐿) = (LSpan‘(ringLMod‘𝐿)) | |
46 | 43, 44, 45 | 3eqtr4g 2669 | . 2 ⊢ (𝜑 → (RSpan‘𝐾) = (RSpan‘𝐿)) |
47 | 38, 46 | jca 553 | 1 ⊢ (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 I cid 4948 ‘cfv 5804 (class class class)co 6549 ndxcnx 15692 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 Scalarcsca 15771 ·𝑠 cvsca 15772 LSubSpclss 18753 LSpanclspn 18792 ringLModcrglmod 18990 LIdealclidl 18991 RSpancrsp 18992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-sca 15784 df-vsca 15785 df-ip 15786 df-lss 18754 df-lsp 18793 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-rsp 18996 |
This theorem is referenced by: crngridl 19059 |
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